**Background**- Collective inertia is strongly influenced at the level crossing at which
quantum system changes diabatically its microscopic configuration. Pairing
correlations tend to make the large-amplitude nuclear collective motion more adiabatic
by reducing the effect of those configuration changes.
Competition between pairing and level crossing is thus expected to
have a profound impact on spontaneous fission lifetimes.
**Purpose**- To elucidate the role of nucleonic pairing on spontaneous fission, we
study the dynamic fission trajectories of Fm and Pu
using the state-of-the-art self-consistent framework.
**Methods**- We employ the superfluid nuclear density functional theory with the
Skyrme energy density functional SkM and a density-dependent
pairing interaction. Along with shape variables, proton and neutron
pairing correlations are taken as collective coordinates. The
collective inertia tensor is calculated within the nonperturbative
cranking approximation. The fission paths are obtained by using the
least action principle in a four-dimensional collective space of
shape and pairing coordinates.
**Results**- Pairing correlations are enhanced along the minimum-action fission
path. For the symmetric fission of Fm, where the effect of
triaxiality on the fission barrier is large, the geometry of fission
pathway in the space of shape degrees of freedom is weakly impacted
by pairing. This is not the case for Pu where pairing
fluctuations restore the axial symmetry of the dynamic fission
trajectory.
**Conclusions**- The minimum-action fission path is strongly impacted by nucleonic
pairing. In some cases, the dynamical coupling between shape and
pairing degrees of freedom can lead to a dramatic departure from the
static picture. Consequently, in the dynamical description of
nuclear fission, particle-particle correlations should be considered
on the same footing as those associated with shape degrees of
freedom.

*Introduction* -- Nuclear fission is a fundamental phenomenon
that is a splendid example of a large-amplitude collective motion
of a system in presence of many-body tunneling. The corresponding
equations involve potential, dissipative, and inertial terms
[1]. The individual-particle motion gives rise to shell
effects that influence the fission barriers and shapes on the way to
fission, and also strongly impact the inertia tensor through the crossings of
single-particle levels and resulting configuration changes
[2,3,4]. The residual interaction between
crossing configurations is strongly affected by nucleonic pairing:
the larger pairing gap the more adiabatic is the collective
motion [5,6,7,8,9,10].

The enhancement of pairing correlations along the fission path was postulated in early Ref. [11] using simple physical arguments. Since the collective inertia roughly depends on pairing gap as [12,5,13,14,15], by choosing a pathway with larger , the fissioning nucleus can lower the collective action. This means that in searching for the least action trajectory the gap parameter should be treated as a dynamical variable. Indeed, macroscopic-microscopic studies [16,17,18,19] demonstrated that pairing fluctuations can significantly reduce the collective action; hence, affect predicted spontaneous fission lifetimes.

Our long-term goal is to describe spontaneous fission (SF) within the superfluid nuclear density functional theory by minimizing the collective action in many-dimensional collective space. The important milestone was a recent paper [20], which demonstrated that predicted SF pathways strongly depend on the choice of the collective inertia and approximations involved in treating level crossings. The main objective of the present work is to elucidate the role of nucleonic pairing on SF by studying the dynamic fission trajectories of Fm and Pu in a four-dimensional collective space. In addition to two quadrupole moments defining the elongation and triaxiality of nuclear shape we consider the strengths of neutron and proton pairing fluctuations. Since in our model the effect of triaxiality on the fission barrier is larger for Fm (4MeV) [21] than for Pu (2MeV) [22], by considering these two cases we can study the interplay between pairing dynamics and symmetry breaking effects [23,9].

*Theoretical framework* --
To calculate the SF half-life, we closely follow the formalism
described in Ref. [20]. In the semi-classical
approximation, the SF half-life can be written
as [24,25]
, where is the
number of assaults on the fission barrier per unit time (here we
adopt the standard value of
) and
is the penetration probability expressed in terms of
the fission action integral,

where , , and represent the HFB hamiltonian, axial () and nonaxial () components of the mass quadrupole moment operator, and neutron () and proton () particle-number operators, respectively. The particle-number dispersion terms , controlled by the Lagrange multipliers , determine dynamic pairing correlations of the system [27,28]. That is, corresponds to the static HFB pairing. Dynamic pairing fluctuations stronger than those obtained within the static solution are described by . The overall magnitude of pairing correlations (static + dynamic) can be related to the average pairing gap [29,30]. In this study, are used as two independent dynamical coordinates to scan over a wide range of pairing correlations (or ).

The one-dimensional path is defined in the multidimensional collective space by specifying the collective variables as functions of path's length . Furthermore, to render collective coordinates dimensionless, we define , where are appropriate scale parameters that are also used when determining numerical derivatives of density matrices [31]. Although the collective action is invariant to uniform scaling, working with dimensionless is simply convenient when defining the fission path and analyzing results. We take b [31], whereas values of MeV were selected after numerical tests of the corresponding derivatives. Namely, we checked that the inertia tensor does not change by increasing these steps up to as large a value as 0.05MeV. Dynamical coordinates and control, respectively, the isoscalar and isovector pairing fluctuations.

As a continuation of our previous study [20], we first consider the SF of Fm, which is predicted to undergo a symmetric split into two doubly magic Sn fragments [32]. Therefore, the crucial shape degrees of freedom in this case are elongation and triaxiality; they are represented by quadrupole moments and defined as in Table 5 of Ref. [33]. To compute the total energy and inertia tensor , we employed the symmetry-unrestricted HFB solver HFODD (v2.49t) [34]. To be consistent with the previous work [20], we use the Skyrme energy density functional SkM [35] in the particle-hole channel.

The particle-particle interaction is approximated by the density-dependent mixed pairing force [36]. The zero-point energy is estimated by using the Gaussian overlap approximation [17,37,38]. To obtain the expression for , we neglected the derivatives of the pairing fields with respect to as we found that the variation of average pairing gap with is quite small. Moreover, we checked that the topology of the fission path is hardly sensitive to the detailed structure of .

The inertia tensor was obtained from the nonperturbative cranking approximation to Adiabatic Time Dependent HFB as described in Refs. [31,20]. The density-matrix derivatives with respect to collective coordinates used to compute [31], were obtained by using finite differences with steps . Finally, to obtain the minimum action pathways we adopted two independent algorithms to ensure the robustness of the result: the dynamical programing method (DPM) [24] and the Ritz method [25]. In all cases considered, both approaches give consistent answers.

*Results* --
In the first step, to assess the
relative importance of isoscalar and isovector pairing degrees of
freedom, the minimum-action path was calculated in the
three-dimensional space of coordinates , , and . To this end, we adopted a 906131 mesh with
,
, and
.
Coordinate was fixed according to the two-dimensional
dynamical path of Ref. [20]. The contour maps of in
the - plane for and - plane for are displayed in
Fig. 1 (left).

Figure 4 summarizes our results for Fm. Namely, it shows , , , and along the fission paths calculated with dynamical (3D) and static (2D) pairing. Compared to the 2D path, the 3D path is shorter and it favors lower collective inertia at a cost of higher potential energy, both being the result of enhanced pairing correlations. It is interesting to notice that the collective potentials in 2D and 3D are fairly different, and they both deviate from the static result that is usually interpreted in terms of a fission barrier, or a saddle point.

While the least-action pathways in Fm are not that far from the static SF path, this is not the case for Pu, where the energy gain on the first barrier resulting from triaxiality is around 2MeV, that is, significantly less than in Fm. To illustrate the impact of pairing fluctuations on the SF of Pu, we consider the least-action collective path between its ground state and superdeformed fission isomer. In this region of collective space, reflection-asymmetric degrees of freedom are less important; hence, the 3D space of () is adequate.

*Conclusions* -- In this study, we extended the self-consistent
least-action approach to the SF by considering collective coordinates
associated with pairing. Our approach takes into account essential
ingredients impacting the SF dynamics [23]: (i) spontaneous
breaking of mean field symmetries; (ii) diabatic configuration
changes due to level crossings; (iii) reduction of nuclear inertia
by pairing; and (iv) dynamical fluctuations governed by the
least-action principle.

We demonstrated that the SF pathways and lifetimes are significantly influenced by the nonperturbative collective inertia and dynamical fluctuations in shape and pairing degrees of freedom. While the reduction of the collective action by pairing fluctuations has been pointed out in earlier works [11,14,16,17,18] and also very recently in a self-consistent approach [39], our work shows that pairing dynamics can profoundly impact penetration probability, that is, effective fission barriers, by restoring symmetries spontaneously broken in a static approach.

Our calculations for Fm and Pu show that the dynamical coupling between shape and pairing degrees of freedom can lead to a dramatic departure from the standard static picture based on saddle points obtained in static mean-field calculations. In particular, for Pu, pairing fluctuations restore the axial symmetry around the fission barrier, which in the static approach is broken spontaneously. The examples presented in this work, in particular in Figs. 4 and 5, illustrate how limited is the notion of fission barrier.

The future improvements, aiming at systematic comparison with experiment, will include: the full Adiabatic Time Dependent HFB treatment of collective inertia, adding reflection asymmetric collective coordinates, and employing energy density functionals optimized for fission [40]. The work along all these lines is in progress.

Discussions with G.F. Bertsch, K. Mazurek, and N. Schunck are gratefully acknowledged. This study was initiated during the Program INT-13-3 ``Quantitative Large Amplitude Shape Dynamics: fission and heavy ion fusion" at the National Institute for Nuclear Theory in Seattle. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under Award Numbers No. DE-FG02-96ER40963 (University of Tennessee) and No. DE-SC0008499 (NUCLEI SciDAC Collaboration); by the NNSA's Stewardship Science Academic Alliances Program under Award No. DE-FG52-09NA29461 (the Stewardship Science Academic Alliances program); by the Academy of Finland and University of Jyväskylä within the FIDIPRO programme; and by the Polish National Science Center under Contract No. 2012/07/B/ST2/03907. An award of computer time was provided by the National Institute for Computational Sciences (NICS) and the Innovative and Novel Computational Impact on Theory and Experiment (INCITE) program using resources of the OLCF facility.

Jacek Dobaczewski 2014-12-06